Experiment #41 · Thought experiment

The Sleeping Beauty Problem

Self-locating probability and Bayesian updating

Adam Elga (modern formulation); origins in Piccione & Rubinstein 1997 · 2000 · Probability, formal epistemology

First published: A. Elga, "Self-locating Belief and the Sleeping Beauty Problem", *Analysis* 60 (2000): 143–147.

Sleeping Beauty is told the protocol, then put to sleep. On waking, what credence should she assign to the fair coin having landed heads — 1/2 or 1/3?

Beauty is told the protocol: a fair coin is flipped. If heads, she is woken once (Monday); if tails, she is woken twice (Monday and Tuesday) with a memory-erasing drug between wakings, so each waking feels the same. On any waking, what should her credence be that the coin came up heads? "Halfers" say 1/2 — the coin was fair and no new information has arrived. "Thirders" say 1/3 — of the three subjectively identical wakings (H-Mon, T-Mon, T-Tue), only one corresponds to heads. The case is the cleanest test of how to update on self-locating information, and it bears directly on anthropic reasoning in cosmology (Doomsday, Boltzmann brains, simulation arguments).

Formulation

Fair coin flipped Sunday; Beauty woken Monday if heads, Monday and Tuesday (with amnesia between) if tails. On waking, Beauty has no temporal information beyond knowing the protocol. Compute P(heads | awoken).

Compare this experiment

Dimensions Engaged

◎

Observer

Targets Observer · Number and Observer · Knowledge Retainment: how should an agent reason when their epistemic situation is duplicated across time?

⧉

Information

A pure test of Information · Granularity: is the self-locating uncertainty additional information that requires updating, or is it captured by the original protocol?

⧖

Time

Engages Time · Direction and self-locating knowledge: how does an agent who has lost track of *when* they are reason about events?

Responses — How Schools Engage

Affirms / takes the bait 2

Simulation Theory

Thirder reasoning generalises directly to simulation arguments: when populations of subjectively indistinguishable observers are weighted, larger populations dominate the prior, with substantive consequences.

Multiverse Theory

Anthropic reasoning in cosmology (Boltzmann brains, Doomsday argument) inherits the same self-locating structure; the Sleeping Beauty debate is the formal core of those disputes.

Denies / rejects the premise 1

Logical Positivism

Where no operational outcome distinguishes the positions, the dispute is verbal. The puzzle is at best a recommendation about probability conventions.

Reframes the question 2

Naturalism

A bet-test (long-run frequency of correct heads-guesses over many repetitions) decisively favours the thirder answer; halfer intuitions trace to a subtle confusion about what is conditionalised on.

Pragmatism

The "right" answer is the one with the best betting behaviour for the question at hand. Halfer and thirder are answering different questions; the apparent contradiction reflects an ambiguity in the setup.

Holds it inconclusive 1

Analytic Metaphysics / Logical Atomism

A genuine live debate: leading thirders include Elga, Bostrom; leading halfers include Lewis (early), White. The disagreement turns on whether self-locating beliefs are subject to Bayesian conditionalisation.